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"... In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable wa ..."

In the &quot;little theories&quot; version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.

...ges of the monoid axioms under I may not be theorems of B. Although I is not properly a theory interpretation, it is what we call a context theory interpretation from M to B relative to the \context&q=-=uot; [22, 10]-=- containing the assumptions f' 1 ; : : : ; ' n g. Context theory interpretations 12 are used in imps in much the same as way as ordinary theory interpretations, so long as our position in a proof lice...

"... : Our ultimate goal is to provide a framework and a methodology which will allow users, and not only system developers, to construct complex reasoning systems by composing existing modules, or to add new modules to existing systems, in a "plug and play" manner. These modules and systems ..."

: Our ultimate goal is to provide a framework and a methodology which will allow users, and not only system developers, to construct complex reasoning systems by composing existing modules, or to add new modules to existing systems, in a &quot;plug and play&quot; manner. These modules and systems might be based on different logics; have different domain models; use different vocabularies and data structures; use different reasoning strategies; and have different interaction capabilities. This paper makes two main contributions towards our goal. First, it proposes a general architecture for a class of reasoning systems called Open Mechanized Reasoning Systems (OMRSs). An OMRS has three components: a reasoning theory component which is the counterpart of the logical notion of formal system, a control component which consists of a set of inference strategies, and an interaction component which provides an OMRS with the capability of interacting with other systems, including OMRSs and hum...

...valid in the system. Notions of a rule following from a set of rules and a rule being derivable from a set of rules are also defined. Reasoning structures generalize the deduction graphs used in IMPS =-=[56, 25]-=- in several ways: a richer domain of sequents; using constraints for provisional reasoning; and nesting. The work presented in this paper is an attempt at an axiomatic presentation of a wide class of ...

"... Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform t ..."

Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform theory which is simultaneously an axiomatic theory and an algorithmic theory. Representing a collection of mathematical models, a biform theory provides a formal context for both deduction and computation. The framework includes facilities for deriving theorems via a mixture of deduction and computation, constructing sound deduction and computation rules, and developing networks of biform theories linked by interpretations. The framework is not tied to a specific underlying logic; indeed, it is intended to be used with several background logics simultaneously. Many of the ideas and mechanisms used in the framework are inspired by the imps Interactive Mathematical Proof System and the Axiom computer algebra system.

...e, fAg is a local context at the position where B occurs in A B and fA 1 ; : : : ; A i 1 g is a local context at the position where A i occurs in ^(A 1 ; : : : ; A n ). The method of local contexts [=-=43]-=- is a powerful idea that is applicable to both deduction and computation. See [28, 30] for examples of how local contexts are used in imps to facilitate deduction and computation. In ffmm, local conte...

"... Chiron is a derivative of von-Neumann-Bernays-Gödel (nbg) set theory that is intended to be a practical, general-purpose logic for mechanizing mathematics. Unlike traditional set theories such as Zermelo-Fraenkel (zf) and nbg, Chiron is equipped with a type system, lambda notation, and definite and ..."

Chiron is a derivative of von-Neumann-Bernays-Gödel (nbg) set theory that is intended to be a practical, general-purpose logic for mechanizing mathematics. Unlike traditional set theories such as Zermelo-Fraenkel (zf) and nbg, Chiron is equipped with a type system, lambda notation, and definite and indefinite description. The type system includes a universal type, dependent types, dependent function types, subtypes, and possibly empty types. Unlike traditional logics such as first-order logic and simple type theory, Chiron admits undefined terms that result, for example, from a function applied to an argument outside its domain or from an improper definite or indefinite description. The most noteworthy part of Chiron is its facility for reasoning about the syntax of expressions. Quotation is used to refer to a set called a construction that represents the syntactic structure of an expression, and evaluation is used to refer to the value of the expression that a construction

...L ker |= ¬free-in(�x�,�A�) when x = z. Hence A is not syntactically closed. A more inclusive definition of free-in could be defined by adding an argument to free-in that represents the local context =-=[17]-=- of its second argument. 957.3 Infinite Dependency 2 Here is another example of a proper expression that depends on the values that are assigned to infinitely many symbols. Let A is the formula infin...

...ur opinion, this corresponds well to the usual mathematical practice.Of course, reasoning inside a formula is not a new idea. To our knowledge, related concepts were first introduced by L.G. Monk in =-=[7]-=- and were further developed in [8]. P.J. Robinson and J. Staples proposed a full-fledged inference system (so called “window inference”) [9] which operated on subexpressions taking the surrounding con...

"... Abstract In the &quot;little theories &quot; version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach ..."

Abstract In the &amp;quot;little theories &amp;quot; version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.

...ges of the monoid axioms under I may not be theorems of B. Although I is not properly a theory interpretation, it is what we call a context theory interpretation from M to B relative to the &quot;context&quot; =-=[22, 10]-=- containing the assumptions f'1; : : : ; 'ng. Context theory interpretations 12sare used in imps in much the same as way as ordinary theory interpretations, so long as our position in a proof licenses...